Equivalent Forms

In a physical piano string, as a specific example, the hammer strikes
the string between its two inputs, some distance from the agraffe and
far from the bridge. This corresponds to the diagram in
Fig.6.15, where the delay lines
are again arranged for clarity of physical interpretation.
Figure 6.15 is almost identical to
Fig.6.14, except that the delay lines now contain samples
of traveling force waves, and the bridge is allowed to vibrate,
resulting in a filtered reflection at the bridge (see
§9.2.1 for a derivation of the bridge filter). The
hammer-string interaction force-pulse is summed into both the left-
and right-going delay lines, corresponding to sending the same pulse
toward both ends of the string from the hammer. Force waves are
discussed further in §C.7.2.

Figure 6.15:
Model of a piano
string struck in its interior by a hammer.

By commutativity of linear, time-invariant elements,
Figure 6.15 can be immediately
simplified to the form shown in
Fig.6.16, in which each
delay line corresponds to the travel time in both directions
on each string segment. From a structural point of view, we have a
conventional filtered delay loop plus a second input which
sums into the loop somewhere inside the delay line. The output is
shown coming from the middle of the larger delay line, which gives
physically correct timing, but in practice, the output can be taken
from anywhere in the feedback loop. It is probably preferable in
practice to take the output from the loop-delay-line input. That way,
other response latencies in the overall system can be compensated.

Figure:
Diagram equivalent to
Fig.6.15, obtained by replacing
the second string input by a separate comb-filter applied to a single
input.

An alternate structure equivalent to
Fig.6.16 is shown in
Fig.6.17, in which the second input
injection is factored out into a separate comb-filtering of the input.
The comb-filter delay equals the delay between the two inputs in
Fig.6.16, and the delay
in the feedback loop equals the sum of both delays in
Fig.6.16. In this case,
the string is modeled using a simple filtered delay loop, and the
striking-force signal is separately filtered by a comb filter
corresponding to the striking-point along the string.